In a two-digit number the tens digit is twice the unit digit. If the digits are interchanged The difference between the original number and new number is twenty-seven. Determine the original and new number.

Let's denote the two-digit number as \(10t + u\), where \(t\) is the tens digit and \(u\) is the units digit. According to the problem, we know that \(t = 2u\). When the digits are interchanged, the new number becomes \(10u + t\). From the given information, we can set up the equation for the difference between the original number and the new number: \[ (10t + u) - (10u + t) = 27 \] Simplifying this gives: \[ 9t - 9u = 27 \] Dividing through by 9 results in: \[ t - u = 3 \] Now, we have two equations: 1. \(t = 2u\) 2. \(t - u = 3\) Substituting the first equation into the second: \[ 2u - u = 3 \] \[ u = 3 \] Now substituting \(u\) back into the first equation to find \(t\): \[ t = 2(3) = 6 \] Thus, the original number is: \[ 10t + u = 10(6) + 3 = 60 + 3 = 63 \] The new number (with digits interchanged) is: \[ 10u + t = 10(3) + 6 = 30 + 6 = 36 \] Therefore, the original number is **63** and the new number is **36**.